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General introduction to
stress reconstruction in
metal plasticitySam Coppieters & Marco Rossi
The mystery of plastic deformation1
1 E.A. Tekkaya, JSTP International Prize Lecture “The mystery of plastic deformation”,
ICTP 2014, Nagoya, Japan
Stan Hoogenboon, Ankara 2002
Experiment
Step 1: Attach weight until the wire elongates plastically
Step 2: Rotate the weight
Experiment: Step 2
Analysis: Step 1 Uniaxial tension causes yielding and plastic elongation of the wire
F
Material element
Movement of dislocations
Uniaxial Tension:
Phenomenological macroscopic response of metals
Dissipation of Energy
Uniaxial Tension:
Phenomenological macroscopic response of metals
Path Dependent
Uniaxial Tension:
Phenomenological macroscopic response of metals
Path Dependent
Constitutive relations in incremental form
Analysis: Step 2
F
T
Ingredient 1: Yield criterion
𝜎𝑒 =1
√2𝜎1 − 𝜎2
2 + 𝜎2 − 𝜎32 + 𝜎3 − 𝜎1
212
where are the dislocations?
𝜎𝑒 =3
2𝝈′: 𝝈′
12
Deviatoric stress tensor = Shear = Slip = movement of dislocations
von Mises (1931)
Ingredient 1: Yield criterion
𝜎𝑒 =1
√2𝜎1 − 𝜎2
2 + 𝜎2 − 𝜎32 + 𝜎3 − 𝜎1
212
where are the dislocations?
𝜎𝑒 =3
2𝝈′: 𝝈′
12
Deviatoric stress tensor = Shear = Slip = movement of dislocations
von Mises (1931)
Ingredient 1: Yield Criterion𝜎𝑒 =
1
√2𝜎11 − 𝜎22
2 + 𝜎22 − 𝜎332 + 𝜎33 − 𝜎11
2 + 6(𝜎122 + 𝜎23
2 + 𝜎312 )
12
𝜎𝑒 =1
√22 𝜎𝑧𝑧
2 + 6 𝜏𝑧θ2 )
12
𝜎𝑌 =1
√22 𝜎2 + 6𝜏2)
12
𝜎
𝜎𝑌
2
+ 3𝜏
𝜎𝑌
2
= 1
Taylor, G.O. and H. Quinney. 1931. The plastic distortion of metals.
Phil. Trans. R. Soc., London A230:323
Ingredient 2: Flow Rule• Ingredient 1 = conditions to initiate yielding
• Ingredient 2 = direction of plastic flow
• Normality hypthesis = associated flow 𝑑𝜺𝑝 = 𝑑𝜆𝜕𝑓
𝜕𝝈
0 30 60 90-45
0
45
90
135DIC
p
0 =
0.002
0.02
0.04
0.08
0.115
0.16
0.20
0.24
0.285
von Mises
Hill '48
Yld2000-2d
M=6
M=7
M=7
y
x
D
irection o
f p
lastic s
train
rate
s
/°
Loading direction /°
𝑑𝜺𝑝
Tangent to the yield surface
𝑑𝜀1𝑝
𝑑𝜀2𝑝
Loading point
𝑑𝜺𝑝 =3
2𝑑𝜀𝑒
𝝈′
𝜎𝑒
Consistency condition
von Mises Material
Normality Hypothesis: Validation1
1 Hakoyama et al., Material Modeling of High Strength Steel Sheet Using Multi-axial Tube Expansion Test with Optical
Strain Measurement System, ESAFORM 2013
0 30 60 90-45
0
45
90
135DIC
p
0 =
0.002
0.02
0.04
0.08
0.115
0.16
0.20
0.24
0.285
von Mises
Hill '48
Yld2000-2d
M=6
M=7
M=7
y
x
Dire
ctio
n o
f p
lastic s
tra
in r
ate
s
/°
Loading direction /°
𝑑𝜀1𝑝
𝑑𝜀2𝑝 𝑑𝜺𝑝
Ingredient 3: Strain hardening law
Assume linear strain hardening (Ingredient 3):
→ 𝑑𝜀𝑒𝑝=𝑑𝜎𝑒𝐻
𝑑𝜺𝑝 =3
2
𝑑𝜎𝑒𝐻𝜎𝑒
𝝈′
von Mises (Ingredient 1) and Flow rule (ingredient 2): 𝑑𝜺𝑝 =3
2𝑑𝜀𝑒
𝑝 𝝈′
𝜎𝑒
𝜎𝑒 = 𝜎0 + 𝐻𝜀𝑒𝑝 Slope H
Analysis: step 1
𝑑𝜀𝑧𝑧𝑝𝑙=3
2
𝑑𝜎𝑒𝐻𝜎𝑒
𝜎′ =3
2
𝑑𝜎𝑒𝐻𝜎𝑒
∙ 𝜎𝑧𝑧 −1
3𝜎𝑧𝑧 =
𝑑𝜎𝑒𝐻𝜎𝑒
∙ 𝜎𝑧𝑧 =𝑑𝜎𝑒𝐻
𝜏𝑧𝜃
𝜎𝑧𝑧A B
Initial yield locus
Subsequent yield locus
Analysis: Step 2
(𝑑𝜀𝑧𝑧𝑝𝑙) 𝑠𝑡𝑒𝑝 2=
𝑑𝜎𝑒𝐻 𝜎𝑒 𝑠𝑡𝑒𝑝 2
∙ 𝜎𝑧𝑧
F
T
𝜎𝑒 𝑠𝑡𝑒𝑝 2 =1
√22 𝜎𝑧𝑧
2 + 6(𝜏𝑧θ2 )
12
𝜎𝑒 𝑠𝑡𝑒𝑝 1 = 𝜎𝑧𝑧
→ 𝑑𝜎𝑒 ≈ 𝜎𝑒 𝑠𝑡𝑒𝑝 2 - 𝜎𝑒 𝑠𝑡𝑒𝑝 1> 0
→ (𝑑𝜀𝑧𝑧𝑝𝑙)𝑠𝑡𝑒𝑝 2 > 0
Analysis: Step 2
𝜏𝑧𝜃
𝜎𝑧A B
Initial yield locus
CF
T
→ (𝑑𝜀𝑧𝑧𝑝𝑙)𝑠𝑡𝑒𝑝 2 > 0
Subsequent yield locus
Computational J2 plasticity
von Mises + strain hardening behavior
=> Process-induced modifications/defects
Distribution of defects
𝑑0 = 0.265 𝑚𝑚
𝜀 ≈ 10−41
𝑠
Cu-wire
Computational J2 plasticity
von Mises + strain hardening behavior: Swift
𝜎𝑒 = 𝐾(𝜀0 + 𝜀𝑒𝑞𝑝𝑙)𝑛
Average behavior
Computational J2 plasticity
von Mises + strain hardening behavior
𝜎𝑒 = 𝐾(𝜀0 + 𝜀𝑒𝑞𝑝𝑙)𝑛
Extrapolation = uncertainty
FE Analysis: Step 2
FE Analysis: improve understandig
Strong 𝜏 −gradient
FE Analysis: support innovation
Industrial application1
1Becker et al., Fundamentals of the incremental tube forming process, CIRP Annals – Manufacturing Technology 63 (2014) 253-256
Reduction of bending moment
HSS
Effort and predictive accuracy
Effort
Effort Accuracy
Experimental
Computational
Financial
Engineering practice: keep things simple
Plasticity Ductile failure criterion
R. von Mises (1913)
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5 Linear
Combined I
Combined II
Combined III
p
p
…but not simpler: Fracture Prediction HET
Hole Expansion Test
High Strength Metal
…but not simpler: Fracture Prediction HET1,2
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5 Linear
Combined I
Combined II
Combined III
p
p
Path dependent
RD
1 Benchmark Problem NUMISHEET 20182 Hakoyama, Nakano and Kuwabara, Fracture Prediction of
Hole Expansion Forming Using Forming Limit Stress Criterion, ESAFORM Conference 2017, Dublin.
Plasticity Ductile failure criterion
Experimental Data
Material modeling(Macroscopic model)
Finite element analysis & Fracture prediction
(Macroscopic model)
General Approach
0 200 400 600 8000
200
400
600
800
y
/ M
Pa
x / MPa
0 :1
2 :1
4 :3
1:13: 41: 2
1: 0
1: 4
4 :1
Stress-controlled material testing*
1st quadrant of stress space
=
17 Mechanical tests
*See presentation prof. Kuwabara
Effort and predictive accuracy
Effort
Effort Accuracy
Minimize time-to-market
Reducing traditional testing by leveraging an existing material test Enhance the synergy between experiments and computational methods
MGI1 = effort to discover, manufacture, and deploy advanced materials
twice as fast, at a fraction of the cost.
Strategic Goal: Integrate Experiments & Computation
1Materials Genome Initiative https://www.mgi.gov/
Experimental Data
Material modeling(Macroscopic model)
Finite element analysis & Fracture prediction
(Macroscopic model)
General Approach
A B
A. Stress-controlled Material Testing
B. Inverse Material Identification
Multi-Scale Approach
Length scale
Meso ContinuumAtomistic …..Quantum
Stru
ctu
ralIn
teg
rityo
f
me
tal c
om
po
ne
nts
& s
tructu
res
Material Characterization
Material Modelling
Simulation & Validation
Design rules/Standardization
Component testingSelection Strategies
Joining
Intrinsic fracture properties Process-induced fracture properties
Electronic
Structure
Molecular
DynamicsDislocation
DynamicsCrystal
Plasticity
0 200 400 600 8000
200
400
600
800
y
/ M
Pa
x / MPa
0 :1
2 :1
4 :3
1:13: 41: 2
1: 0
1: 4
4 :1
Mircostructurally-informed constitutive modeling1
Virtual Material testing
1st quadrant of stress space
=
17 Virtual tests
=
1 day of experimental effort
1 day of computational effort
Microstructure-based plasticity model(Virtual material test)
Material modeling(Macroscopic model)
Finite element analysis(Macroscopic model) -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
0
100
200
300
400
500
600
x:y = 4:1
SPCD
Tru
e s
tress
/ M
Pa
Logarithmic plastic strain p
Cruciform
Tubular x
Tubular y
CP
Mircostructurally-informed constitutive modeling1
Microstructure-based plasticity model(Virtual material test)
Material modeling(Macroscopic model)
Finite element analysis(Macroscopic model)
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.60
100
200
300
400
500
600
x:y = 4:1
SPCD
Tru
e s
tress
/ M
Pa
Logarithmic plastic strain p
Cruciform
Tubular x
Tubular y
CP
Mircostructurally-informed constitutive modeling
Crystal plasticity simulation(Virtual tensile test)
Material modeling(Macroscopic model)
Finite element analysis(Macroscopic model)
0 200 400 6000
200
400
600 CP(ASR) DH
SPCD
y /
MP
ax / MPa
p
0 =
0.002
0.005
0.01
0.03
0.05
0.10
0.15
0.20
0.25
0.289
Mircostructurally-informed constitutive modeling
Crystal plasticity simulation(Virtual tensile test)
Material modeling(Macroscopic model)
Finite element analysis & Fracture prediction
(Macroscopic model)
Mircostructurally-informed constitutive modeling
Experimental Data
Material modeling(Macroscopic model)
Finite element analysis & Fracture prediction
(Macroscopic model)
General Approach
A B C
A. Stress-controlled Material Testing
B. Inverse Material Identification
C. Virtual Material Testing
Plastic material response
=
the result of synergistic effects
DC05
Loading conditions
Environmental conditions
Microstructure
…
Synergistic effects during diffuse necking
Large plastic strains,
Biaxial stress state
& strain path change
Strain rate & Temperature increase
Damage evolution
Plastic material response can be complex…
Differential work hardening
Bauschinger effect
Strain rate sensitivity
Kinematic hardening
Ductile Damage
…
Plastic material response can be complex…
Differential work hardening
Bauschinger effect
Strain rate sensitivity
Kinematic hardening
Ductile Damage
…
Isotropic work hardening
𝜎
𝜀
Differential work hardening
Experimental Data
Material modeling(Macroscopic model)
Finite element analysis (Macroscopic model)
Problem statement
Problems
Many models exist
Anisotropic Yield Functions
Hill 1948
Hill 1990
Gotoh
Barlat Yld2000-2d
Barlat Yld2004
Yoshida
BBC2005
BBC2008
Vegter Spline
Karafills-Boyce
CPB2006
…
Coupled Damage Models
Experimental Data
Material modeling(Macroscopic model)
Finite element analysis (Macroscopic model)
Problem statement
Selection strategy
Identification strategy
Problems
Many models
Need
Experimental Data
Material modeling(Macroscopic model)
Finite element analysis (Macroscopic model)
Problem statement
(Selection strategy)
Identification strategy
Problems
Many models
Need
Models not available
in commercial FE code Implementation
Experimental Data
Material modeling(Macroscopic model)
Finite element analysis (Macroscopic model)
Problem statement
(Selection strategy)
Identification strategy
Problems
Many models
Need
Models not available
in commercial FE code Implementation
Implementation = stress reconstruction
FE CODE
Stress integration – Stress reconstruction -Stress update
Stress update algorithm
Experimental Data
Material modeling(Macroscopic model)
Finite element analysis (Macroscopic model)
Problem statement
(Selection strategy)
Identification strategy
Problems
Many models
Need
Models not available
in commercial FE code Implementation
Many codes
Unified Material Model Driver for plasticity1
1JANCAE, The Japan Association for Nonlinear CAE
Experimental Data
Material modeling(Macroscopic model)
Finite element analysis (Macroscopic model)
Problem statement
(Selection strategy)
Identification strategy
Problems
Many models
Need
Models not available
in commercial FE code Implementation
Many codes
Inverse Material Identification
Numerically converged value (e.g. FEMU)
Experimentally measured value (e.g. VFM)
Finite Element Model Updating (FEMU)
Finite Element Model Updating (FEMU)
Virtual Fields Method
Global Equilibrium
Energy Method
V (DIC)
y
xx
y
Energy MethodDIC
(U,V)STRAIN RECONSTRUCTION STRESS RECONSTRUCTION
LE22 S22
∆𝜀𝑖
Converged steps in FEA
During a quasi-static test the internal work equals the external work
Lines A-B and C-D remain straight The Volume V is constant The stresses and strains are constant
through the thickness of the sheet
y
x
𝝈𝒊𝒋 , 𝜺𝒊𝒋 V, S, t
A B
C D
𝑓𝑖 , 𝑢𝑖
Energy Method
Forming Limit Curve
Forming Limit Curve*
* Junying Min et al., Compensation for process-dependent effects in the determination of localized necking limits,
International Journal of Mechanical Sciences117(2016)115–134
Compensate for nonlinear strain path